Collaborative Research: OAC Core: Robust, Scalable, and Practical Low Rank Approximation

近年来，数据以更快的速度产生，几乎影响到社会的各个方面。来自实验、观察和模拟的数据不仅在更经典的科学和工程领域，而且在许多其他领域，如商业跟踪越来越多的消费者行为方面，以及社交网络捕获大量关于人与人之间关系及其行为和互动的信息。迫切需要将一组数据提炼成一个较小的表示形式，将有用的信息从噪声中分离出来，并捕获最重要的趋势、模式和潜在的关系。 这种表示可以用于直接解释隐藏的模式，或者作为简化其他数据分析任务的一种手段。 这个项目通过研究线性代数中的一个叫做低秩近似的概念来解决这些挑战。 该项目开发了忠实地从数据集中提取有意义信息的技术。 这些算法还设计用于利用高性能计算机，以便分析师能够更快地获得结果并解决更大的问题。 该项目的总体努力有望缩小能够有效处理大规模问题的算法与数据分析师将原始输入转换为有意义的表示和可操作的见解的能力之间的差距。在这个项目中研究的矩阵和张量低秩近似是许多科学和工程应用的基础工具。在低秩近似上施加约束可以对许多关键问题进行建模，并且设计可扩展的算法可以实现远远超出经典科学和工程学科的新应用。特别是，具有非负数据值的数学模型比比皆是，并且施加非负约束允许更精确和可解释的模型。可以设计这些约束的变体来反映现实数据分析问题的其他特征。该项目的主要目标是(1)开发用于评估用于等级和模型确定的计算低秩近似的稳健技术，(2)在当今的极端规模机器上开发可扩展的并行算法，用于大型和稳健的低秩近似，以及(3)为最终用户提供大规模计算和分析解决方案所需的实用工具。典型的数据和应用科学家使用Python或Matlab来迭代计算、可视化和评估解决方案，并且它们仅限于具有可行内存和计算需求的小数据集。虽然存在高性能算法和实现，但如果最终用户不能依赖结果的健壮性和泛化性，他们就不会利用这些工具。该项目旨在缩小这一差距，为数据分析工作流程的所有步骤开发一个具有可扩展解决方案的端到端系统。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Nearly all aspects of society are affected by data being produced at a faster rate in recent years. The data from experiments, observations, and simulations are not only in more classical science and engineering domains but also in numerous other areas such as businesses tracking more and more facets of consumer behavior, and social networking capturing vast amounts of information on the relationships between people and their actions and interactions. There is a strong need to distill a set of data into a smaller representation that separates useful information from noise and captures the most important trends, patterns, and underlying relationships.  Such a representation can be used for direct interpretation of hidden patterns or as a means of simplifying other data analytic tasks.  This project addresses these challenges by studying a concept from linear algebra called low rank approximation.  The project develops techniques that faithfully distill the meaningful information within a data set.  The algorithms are also designed to exploit high-performance computers so that analysts can get results more quickly and tackle larger problems.  The overall effort in the project is expected to close the gap between algorithms that can effectively handle very large-scale problems and the data analyst’s ability to convert raw input into meaningful representations and actionable insight.The matrix and tensor low rank approximations being studied in this project serve as foundational tools in numerous science and engineering applications. Imposing constraints on the low rank approximations enables the modeling of many key problems, and designing scalable algorithms enables new applications that reach far beyond classical science and engineering disciplines. In particular, mathematical models with nonnegative data values abound, and imposing nonnegative constraints allows for more accurate and interpretable models. Variants of these constraints can be designed to reflect additional characteristics of real-life data analytics problems. The primary goals of this project are (1) to develop robust techniques for evaluating computed low rank approximations for rank and model determination, (2) to develop scalable parallel algorithms for large and robust low rank approximations on today’s extreme-scale machines, and (3) to provide end users the practical tools required to compute and analyze solutions at scale. Typical data and application scientists use Python or Matlab to iteratively compute, visualize, and evaluate solutions, and they are limited to small data sets with feasible memory and computational requirements. While high-performance algorithms and implementations exist, end users would not leverage these tools if they cannot rely on the robustness and generalizability of the results. This project aims to close this gap, developing an end-to-end system with scalable solutions for all steps of the data analytics workflow.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.